The mergegram of a dendrogram and its stability

This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay

Inhibition-augmented trainable COSFIRE filters for keypoint detection and object recognition

The shape and meaning of an object can radically change with the addition of one or more contour parts. For instance, a T-junction can become a crossover. We extend the COSFIRE trainable filter approach which uses a positive prototype pattern for configuration by adding a set of negative prototype patterns. The configured filter responds to patterns that are similar to the positive prototype but not to any of the negative prototypes. The configuration of such a filter comprises selecting given channels of a bank of Gabor filters that provide excitatory or inhibitory input and determining certain blur and shift parameters. We compute the response of such a filter as the excitatory input minus a fraction of the maximum of inhibitory inputs. We use three applications to demonstrate the effectiveness of inhibition: the exclusive detection of vascular bifurcations (i.e., without crossovers) in retinal fundus images (DRIVE data set), the recognition of architectural and electrical symbols (GREC’11 data set) and the recognition of handwritten digits (MNIST data set)